# Thread: Procedure to Get Derivative of exp(x) Respect to x

1. Excuse me ... .

The definition of derivative a function respect to is

... .

Consider ... then

... .

In real calculus, we have known that ... .

So, we have to say that

... (*) ... .

The problem is how to prove the last equality (*) ... .

Maybe, we can prove the last equality by L'Hopital Theorem ... .

But, the L' Hopital Theorem uses the concept of derivative ... .

Whereas, the concept of derivative uses the concept of limit ... .

How can the last equality (*) be proven without L'Hopital Theorem ... ?

Is the last equation (*) an assumption ... ?

Thank you very much ... .

2.

3. The demonstration depends on the definition that you accept for exp(x) :
• my favorite definition is that exp(x) is a function such as it is equal to its derivative. .
• You can also define exp(x) as the inverse function of Log. The demo is easy using formula of the derivative of inverse function :
. Check the conditions required before using this formula.

4. You could use the power series for e^x.

5. Excuse me ... .

I have known that the definition of Euler number is ... .

If we use the power series procedure, we must use concept of derivative ... because the in Maclaurin series exist derivative expression, that is

... .

Using Maclaurin series, we have to seek derivative of ... , whereas we will prove that derivative of is ... .

I'm sorry if I do mistakes ... . Thank you for the attention ... .

6. As suggested by mathman, another possible definition for exp(x) is the limit of the series

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